You can theoretically clear screen burn by "reverse burning" the monitor with a pattern that exactly matches the opposite of what's burned in. In practice, doing so is impossible: you'll just end up with an even more objectionable pattern since you won't get it lined up perfectly, and it would take forever anyway. Either live with it, replace the monitor, or swap tubes.
Solid colors or "static" won't help. All it will really do is evenly wear the whole CRT even further resulting in a dimmer display that still has about the same level of burn.
Plasmas can be very susceptible to burn, even more so than direct view CRT. Early sets especially were bad. Modern sets have better pixel oribters and some even share the gasses between multiple cells. It's still quite possible to burn them, though, especially in high-duty monitor applications like arcade games. Every arcade game I've seen with a Plasma, even modern ones that are less than a year old, have pretty noticeable burn-in.
"Static" can help eliminate non-permanent "retention" on plasma displays. I don't recall the physics behind it, but it does work. Permanent burn (phosphor degradation) won't be fixed by this.
Rear-projection CRT is probably the worst when it comes to burn. Fortunately, these aren't very common anymore.
LCD is not generally susceptible to burn-in. In extreme cases, UV from the CCFL backlight can slightly discolor the polarizer on the front in white spots. Improperly driven LCDs can sustain a burn-like effect to the panel itself due to crystal damage. This is especially common on STN gray displays like supermarket credit-card terminals, but most TVs and computer monitors seem to get it reasonably right, though some models have excessive overdrive in order to attain quick response time to eliminate ghosting (these also typically have lots of lag).
For home use, if you set the display properly, you won't burn it any time in the next several (dozen, probably) years. If you crank the contrast sky-high (sadly, this is often the default), you might get some problems if you use it a lot.